MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax-addcl Unicode version

Axiom ax-addcl 8884
Description: Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by theorem axaddcl 8860. Proofs should normally use addcl 8906 instead, which asserts the same thing but follows our naming conventions for closures. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
Assertion
Ref Expression
ax-addcl  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )

Detailed syntax breakdown of Axiom ax-addcl
StepHypRef Expression
1 cA . . . 4  class  A
2 cc 8822 . . . 4  class  CC
31, 2wcel 1710 . . 3  wff  A  e.  CC
4 cB . . . 4  class  B
54, 2wcel 1710 . . 3  wff  B  e.  CC
63, 5wa 358 . 2  wff  ( A  e.  CC  /\  B  e.  CC )
7 caddc 8827 . . . 4  class  +
81, 4, 7co 5942 . . 3  class  ( A  +  B )
98, 2wcel 1710 . 2  wff  ( A  +  B )  e.  CC
106, 9wi 4 1  wff  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
Colors of variables: wff set class
This axiom is referenced by:  addcl  8906
  Copyright terms: Public domain W3C validator